Son harmonique

\[ x(t) = \sum\limits_{k\in\mathbb{N}} A_k\cdot\cos(2\pi k f_0 t) \]

Fourier

\[\hat{x}(f) = \int\limits_{-\infty}^{\infty} x(t)\cdot e^{-2\pi j ft}\mathrm{d}t\]

Transformée de Fourier à court-terme (STFT)

\[X(t, f) = \int\limits_{-\infty}^{\infty} x(\tau)\cdot w(\tau-t)\cdot e^{-2\pi j f\tau} \mathrm{d}\tau \]

Transformées discrètes:

DFT:

\[ X[k] = \sum\limits_{n=0}^{N-1} x[n]\cdot e^{-2\pi j k \frac{n}{f_s}}\]

STFT:

\[X[n, k] = \sum\limits_{n=0}^{N-1} x[m]\cdot w[m-n]\cdot e^{-2\pi j k \frac{m}{f_s}}\]

\[w[n] = \sin^2\left(\frac{\pi n}{N -1}\right) \]